Setup for Problems 3 and 4. Both of these problems are about a iden- tical particles confined to a one-dimensional box. Specifically. N identical particles of mass m are confined by a box (infinite square well) to the region 0

(20pts) Fermions in a 1D box. to spin,

(a) Show that the density of one-particle momentum states for this system is of the form D(E) = BLE- and compute the constant B.

(b) N identical spin-? fermions are in this 1D box. Compute the Fermi energy Er (at T = 0). Is Er intensive or extensive? Justify your answer from your expression for Er.

4. (10pts) Bosons in a 1D box. Now V identical spin-0 bosons are in this box. When this system is cooled down, is there a BEC? For credit you must justify your answer,

(a) Show that the density of one-particle momentum states for this system is of the form D()=BL1/2 and compute the constant B. (b) N identical spin- 21 fermions are in this 1D box. Compute the Fermi energy F (at T=0 ). Is F intensive or extensive? Justify your answer from your expression for F. 1. (10pts) Bosons in a 1D box. Now N identical spin-0 bosons are in this box. When this system is cooled down, is there a BEC? For credit you must justify your